I'm currently learning about directional derivative, and I need to figure out something in order to fully understand it: What I understood is that directional derivatives are about infinitesimal changes in directions, then we are not really interested in the magnitude of that change.
So, can we say that the value of the directional derivative of the vector V and the vector W = 100V (for example) at the point (x,y) are supposed to be almost equal?
On
[ Made my comment an answer at the encouragement of @amd - thanks!]
We are interested in the magnitude of the change. The directional derivative is the rate of change of the function in a particular direction (specified by a unit vector in that direction). If you take a function of two variables and think about its graph (a surface in 3-space), then pick a direction in the x-y plane and cut the graph with a vertical plane in that direction, the intersection of the plane and the graph is a planar curve, and then you can pretend that everything happens in that plane and do ordinary one-dimensional calculus there.
For any $\vec v=(a,b)$ we can define the directional derivative as:
$$\frac{\partial f}{\partial \vec v}=\lim_{h\to 0}\frac{f(x_0+ah,y_0+bh)-f(x_0,y_0)}{h}$$
and if we consider the corresponding unit vector $\hat v=(c,d)$ such that $\vec v = \lambda \hat v$ we have
$$\frac{\partial f}{\partial \vec v}=\frac{\partial f}{\partial \lambda \hat v}=\lim_{h\to 0}\lambda\frac{f(x_0+\lambda ch,y_0+\lambda dh)-f(x_0,y_0)}{\lambda h}=\lambda \frac{\partial f}{\partial \hat v}$$